FOM: Do We Need New Axioms? Upcoming Panel Discussion

[Harvey Friedman]

"Do We Need New Axioms?" is the title of a panel discussion between Sol
Feferman, Harvey Friedman, Penelope Maddy, John Steel at the Annual Meeting
of the ASL, June  3-7, 2000, in Urbana.

This is the first of a two part posting. The second posting follows, and
presents relevant new results and conjectures.

All four of us have agreed to send each other drafts of our views to each
other, as we intend to present them, before the meeting. We are not
scheduled to do this yet, but my very latest research bears directly on
this topic. So I will start a discussion of this topic here on the FOM now.

Feedback from the FOM will help air the crucial issues more thoroughly than
otherwise before the actual panel discussion takes place.

I have talked to all of these panelists about their positions on these
matters over the years, and I think I have at least a superficial
understanding of their views. They will, of course, speak eloquently for
themselves. In fact, they represent four distinct views - or at least four
views with distinct emphases.

I will now give a brief account of my understanding of the four views. I
will follow this by a second round of accounts.

I have sent a copy of these two postings to Feferman, Maddy, and Steel.

1. Maddy. My impression of her view is this. There is an obviously
fundamental problem in set theory of great interest in and out of
mathematics, namely the continuum hypothesis. It is known that we cannot
settle the continuum hypothesis using the usual axioms for mathematics as
represented by ZFC - unless those axioms are inconsistent. Since ZFC is
consistent, we therefore need new axioms to settle the continuum
hypothesis.

We can go further with this line of reasoning even under the unlikely event
that ZFC turns out to be inconsistent. For then ZFC will be replaced by a
suitable fragment of ZFC that is suitably adequate for mathematical
practice and believed to be consistent. But the continuum hypothesis will
then re-emerge as neither provable nor refutable in that fragment. Thus we
will wind up in the same situation. For instance, if an inconsistency is
found in ZFC using Replacement, then ZC = Zermelo set theory with the axiom
of choice may well emerge as the standard axioms for mathematics. Then we
will still need new axioms to resolve the continuum hypothesis, unless even
that fragment is inconsistent. This process may conceivably repeat itself
many times, but in any case we will always have to search for new axioms to
settle the continuum hypothesis.

MY DIGRESSION: An interesting technical problem emerges naturally along
this line of reasoning. Namely, Godel and Cohen showed that the continuum
hypothesis is neither provable nor refutable in ZFC under the assumption
that ZFC is consistent, using only very low level arithmetical reasoning.
For which fragments of T does this situation prevail? I.e., that one can
show that the continuum hypothesis is neither provable nor refutable in T
under the assumption that T is consistent, using only very low level
arithmetical reasoning? This situation seems to prevail for any "natural"
T, but one can investigate just what appropriate conditions need to be
placed on T in order for this to hold. END OF DIGRESSION

Maddy is aware that many mathematicians are not interested in the continuum
hypothesis. However, Maddy would emphasize the overriding fundamental
nature of the continuum hypothesis, and thereby give a strong affirmative
answer to "Do We Need New Axioms?"

As a "naturalist", Maddy would be expected to be interested in taking into
account

*identifiable reasons why mathematicians are not interested in the
continuum hypothesis*

in her view on these matters. Working primarily in the philosophy
community, Maddy is unlikely to have had detailed interactions with
mathematicians who are not interested in the continuum hypothesis, and so
probably has not worked the disinterest of the mathematicians in the
continuum hypothesis into her views yet. However, Maddy might be interested
in doing this in connection with this upcoming panel discussion.

2. Steel. My impression of his view is this. Steel would agree with all of
what I attributed above to Maddy except for the last paragraph. In
addition, Steel would emphasize the large cardinal axioms as what he views
is the canoncial extension of the usual ZFC axioms, and their productive
use in settling problems about the projective sets.

In particular, Steel is aware of the disinterest among mathematicians in
the continuum hypothesis and the projective hierarchy. But, as opposed to
Maddy, for Steel this is not an intellectual matter worthy of research
investigation that is to be worked into one's philosophical/mathematical
views or one's research.

More specifically, Steel views the mathematicians' interest/disinterest or
attitudes towards problems and topics in set theory as sociology, which is
of significance only in the role that it plays in funding and job
opportunities. For Steel, this is something that is subject to
unpredictable change and fashion and has no basis in real philosophical or
mathematical issues.

In addition, my impression is that Steel feels that one cannot tell what
mathematics will be important in the future, and thus it is unreasonable,
unproductive, unfair, and irresponsible to criticize work in set theory on
the grounds of lack of direct connection with matters of current interest
to the mathematics community.

It is also my impression that Steel feels that current mainline research in
set theory is based on views that are philosophically attractive, but
perhaps not fully coherent, and certainly not explainable in elementary
terms that are readily accessible to outsiders, even within the
mathematical logic community. But in his view, coherence and explainability
should in no way influence the direction and emphasis of research in
mainline set theory, nor deter or slow down its intensity. In his view, it
is certainly not appropriate to consider coherence and explainability in
the evaluation of research in mainline set theory.

3. Feferman. It is my impression that Feferman disagrees with virtually
every view of Maddy and Steel on these matters. My impression is that
Feferman is not only aware of the disinterest of the mathematicians in the
continuum hypothesis and the projective hierarchy, but feels that their
attitude is fully justified in that the continuum hypothesis in particular
is not clearly meaningful. Feferman has, in fact, come at least close to
stating that it is meaningless. Specifically, he regards it as of such a
fundamentally different character than any currently celebrated
mathematical conjecture as to cause its nonprovability and nonrefutability
from ZFC to be irrelevant to the appropriate meaning of the question "Do We
Need New Axioms?"

I am not sure what Feferman's view is with regard to the meaningfullness or
meaninglessness of standard statements about the projective hierarchy such
as the Lebesgue measurability of projective sets and the like. Such
statements have been shown to follow from large cardinals, whereas it is
already clear from Godel that such statements are refutable from the axiom
of constructibility. I don't know if Feferman puts such statements in the
same category as the continuum hypothesis.

Nevertheless, it is my impression that Feferman also regards these as so
fundamentally different in character than currently celebrated mathematical
conjectures, that their nonprovability and nonrefutability from ZFC does
not bear on the appropriate meaning of the question "Do We Need New Axioms?"

It is also my impression that in Feferman's view, philosophical coherence,
explainability, and direct connections with matters of current interest to
the mathematics community must be crucial issues for mainline set theory
research. It is my impression that for Feferman, they must play a role in
the direction and evaluation of that research. And it is my impression that
on these grounds, Feferman believes that current research in mainline set
theory is deficient.

It is my impression that Feferman does not recognize any canonical nature
of the large cardinal hierarchy as the unique extension and extrapolation
of the ideas behind the axioms of ZFC. Nor does Feferman find any reaons
put forth by the set theory community that large cardinals exist or are
consistent compelling.

In fact, it is my impression that Feferman goes much further. It is my
impression that Feferman dismisses the ongoing efforts to uncover natural
questions in discrete mathematics that are neither provable nor refutable
in ZFC  on two grounds.

Firstly, the questions in discrete mathematics that are dealt with in this
work are, as I understand Feferman, not sufficiently natural in order to
bear on the question "Do We Need New Axioms?"

Secondly, the questions in discrete mathematics are proved using certain
large cardinal axioms. But Feferman is not convinced that these large
cardinal axioms are true, or even consistent. In fact, Feferman raises the
possibility that these questions in discrete mathematics are outright
refutable with no additional axioms (which would be the case if these large
cardinal axioms are inconsistent). Therefore, as I understand it, Feferman
concludes that these examples, even if they were sufficiently natural -
which in his view they are not - would not clearly answer the question "Do
We Need New Axioms?" in the affirmative.

4. Friedman. Friedman disagrees sharply with some of Steel's and Feferman's
views. Friedman has not taken a definite position on the meaningfulness
issue with regard to the continuum hypothesis and statements about the
projective hierarchy. However, Friedman has come to the definite conclusion
that we do need new axioms in a very strong sense, based on new results and
conjectures that are put forth in the next posting. So these meaningfulness
issues can be skirted for Friedman's positive answer to the question "Do We
Need New Axioms?"

Friedman believes that a naturalistic investigation in the sense of Maddy
would reveal the reasons behind the disinterest in the continuum hypothesis
and the projective hierarchy among mathematicians. This is the systematic
and fundamental difference in the nature of the objects under discussion.
With the continuum hypothesis, this is rather obvious because of the
essential role of unrestricted sets of real numbers. The great generality
of unrestricted sets of real numbers is what causes the difficulties in
settling the continuum hypothesis.

However, with regard to the projective hierarchy, the situation is more
subtle. This is because the plausible defense can be mounted that any
statement about the projective hierarchy - or at least up to any level of
the projective hierarchy - can be written as a statement that quantifies
only over real numbers. And real numbers are generally essential
unquestioned objects for mathematicians. This defense can be bolstered be
being more specific. Any statement about the projective hierarchy up to any
level can be written as a first order statement in the natural structure of
the ordered field of real numbers together with the predicate "being an
integer." I.e., (R,<,+,*,Z). So why shouldn't mathematicians be interested
in statements about the projective hierarchy, at least up to any level?

However, the flaw in this reasoning is that if a simple statement such as
"every proejctive set of level 3 is Lebesgue measurable" is formulated as a
first order statement in the structure (R,<,+,*,Z) then it becomes
hopelessley unnatural, ugly, and incomprehensible.

So why not state "all projective sets are Lebesgue measurable" - which is
neither provable nor refutable in ZFC - in the equivalent form "all first
order definable sets in the structure (R,<,+,*,Z) are Lebesgue measurable"
to the mathematicians? After all, the mathematicians are interested in the
structure (R,<,+,*,Z). The problem is that mathematicians are not
interested in sets definable by arbitrary first order formulas in even
those structures that they are interested in. They will inevitably evaluate
the interest in this for them according to the extent that they feel
comfortable with the level of generality and pathology. As soon as they get
a whiff that pathology is an underlying issue, they will lose interest.
That is why they are not interested in Lebesgue measurability per se. If,
in a context, Lebesgue measurability is not immediately clear, then they
feel that they are in a context with pathology and uncharacteristic
generality, and they will lose interest.

So where are they comfortable, in terms of no pathology? Certainly in the
integers. They do not speak of "pathological integers". Same with the real
numbers, more or less. And even more comfortable with "algebraic integers."
What about sets of integers? Well, that is a very interesting question. In
general, this is OK. They are kind of stuck with it, because, e.g., the
sets of positive integers are so easily identified with the base two
expansions of their beloved real numbers. Functions on the integers of
several variables? Yes, in general OK. Of course, even clearer is finite
sets of integers, rational numbers, algebraic numbers, and functions of
several variables on the integers with finite domain.

Of course, it is also a matter of what you do with these objects. Comparing
sets of integers by inclusion, or using Boolean operations on them, is
fine. Also, taking forward and inverse images of sets of integers under
functions is generally OK. Combinations of such processes - perhaps thin
ice.

But there must also be a well motivated purpose to the constructions. Not
an encoding of something exotic into such objects for the purpose of
admitting things that are meant to be pathological.

Probably according to Maddy as "naturalist", considerations such as these
are objective facts about the nature of the mathematical enterprise that
ought to be taken into account. Especially in an issue such as "Do We Need
New Axioms?"

However, probably according to Steel, such considerations are irrelevant
sociology of no intellectual significance. Steel is probably convinced by
the argument that once one admits the structure (R,<,+,*,Z) - which the
bulk of mathematicians do - then one is home free at least with regard to
the projective hierarchy. Perhaps not quite home free with regard to the
continuum hypothesis, though. And the reason one is home free with regard
to the projective hierarchy, probably according to Steel, is that the
projective sets are exactly the first order definable sets in that
structure. And, probably according to Steel, the first order definable sets
are unavoidable becuase mathematicians use quantifiers over the reals and
integers, and connectives, and of course <,+,*, and so they have first
order definable sets.

But the problem is that they don't arbitrarily combine these constructions.
That's what you need in order to generate the definable sets. Well, not
exactly arbitrarily, of course. Its better than that because of the normal
form theorems that are relevant here. But nevertheless, serious strings of
alternating quantifiers, which are forward images alternating with
complementation, and one has to use multi-dimensional relations also.
Etcetera.

So when you finally explain what, say, the Lebesgue measurability of
projective sets really means, you don't get the burning interest that you
get with, say, standard conjectures in standard mathematics. And this is
because the essence of the proejctive hierarchy investigations is the
construction of the objects, where algebraic and geometric aspects are
completely blurred or destroyed. Such constructions are not at the heart of
the celebrated conjectures of mathematics.

Here Maddy might carefully consider what is going on here from the point of
view of the mathematicians, and Steel would probably dismiss this as
sociology.

Feferman probably takes such considerations as an essential feature of the
matheamtical enterprise, and uses it as a basis for his negative assessment
of the efforts to construct problems in discrete mathematics which cannot
be settled within ZFC.

Friedman has taken such considerations seriously as a principal motivating
force behind his research for over 30 years. Friedman has insisted that the
emergence of totally convincing problems in discrete mathematics that
cannot be settled within ZFC is an evolutionary process that is well
controlled and well underway, and will definitely succeed.

Friedman has suggested privately to Feferman that it may not be appropriate
to publicly criticize an ongoing research project for failing to achieve
its ultimate goal while it is clearly making progress. That Feferman should
wait until persistent progress has stopped or slowed down.

It is my impression that Feferman may think that there is an intrinsic
reason why such a project will not or can not succeed. That somehow large
cardinals simply do not have any real application in genuine clearly
motivated completely natural elementary discrete mathematical contexts. And
that that is sufficient justification for taking the very unusual step (in
the mathematical community) of publicly criticizing a substantial, deep,
ongoing, intense investigation while substantial progress is being made.

In fact, some such justification is likely, as such public criticism has
the likely effect of discouraging efforts along these lines, as well as
diminishing interest of others in the progress being made.

Feferman has raised the issue about derivations from large cardinals that I
mentioned above in this posting, in the last paragraph under Feferman. But
this is not a genuine issue. Experience with these results show that one
can always naturally weaken the independent statements so that their
logical strength hits a given level of set theory more or less on the
button.

Specifically, let us say that one has a natural discrete problem P that is
equivalent to the 1-consistency of a level A of the large cardinal
hierarchy. And suppose that one is doubtful of the 1-consistency of level
A. And then suppose that one is comfortable with a lower level B of the
large cardinal hierarchy. E.g., A might be Mahlo cardinals of infinite
order, and B might be "infinitely many inaccessible cardinals." Then one
can naturally adjust P - through introduction of relevant natural number
parameters and the like - so that P' is equivalent to the 1-consistency of
level B. Then because of the confidence in level B of the large cardinal
hierarchy, one is convinced that P' represents a necessary use of new
axioms beyond ZFC.

Or, say, one is only comfortable with the level surrounding ZFC. Then the
appropriate weakening P'' can surely be found so that P'' is equivalent to
the 1-consistency of ZFC, and in particular, P'' is provable in MKC but not
in ZFC. And similar results can be obtained corresponding to whatever level
of set theory one is comfortable with - e.g., maybe only ZC = Zermelo set
theory with choice.

5. Maddy. How does one view the existing use of large cardinals in the
projective hierarchy? It is well known that basic facts about the
projective hierarchy such as Lebesgue measurability cannot be settled
within ZFC. As indicated above, this is probably enough for Maddy to
conclude that we need new axioms in the sense she deems most appropriate -
unless she is being carefully naturalistic; certainly enough for Steel to
conclude that we need new axioms in the sense he deems most appropriate;
not enough for Feferman to conclude that we need new axioms in the sense he
deems most appropriate; and not enough for Friedman to conclude we need new
axioms in the sense he deems most appropriate. And Feferman thinks that we
do not need new axioms in the sense he deems most appropriate - and perhaps
thinks that we never will need new axioms in the sense he deems most
appropriate; whereas Friedman thinks that we need new axioms in the sense
he deems most appropriate, and that he is on the brink of definitively
establishing this.

However, the existing use of large cardinals in the projective hierarchy
raises some other issues. Is it a reason to

i) decide that these conclusions in the projective hierarchy are true?
ii) accept these large cardinals?

The complicating issue is that these conclusions are refutable using axioms
of set theory other than large cardinals. Namely, the axiom of
constructibility.

The mainline set theorists argue that large cardinals are good axioms and
the axiom of constructibility is bad. Maddy has tried to formalize this
view, in terms of a principle she calls MAXIMIZE.

The trouble is that there is a certain amount of philosophical incoherence
in all this, because the most obvious formulations of MAXIMIZE don't work,
either because they are inconsistent or don't distinguish properly between
the axiom of constructibility and the large cardinal axioms.

**********

I'll stop here, not because I don't have much more to say, but because this
should be sufficient to generate quite a discussion on the FOM.